891 research outputs found
TheoryGuru: A Mathematica Package to Apply Quantifier Elimination Technology to Economics
We consider the use of Quantifier Elimination (QE) technology for automated
reasoning in economics. There is a great body of work considering QE
applications in science and engineering but we demonstrate here that it also
has use in the social sciences. We explain how many suggested theorems in
economics could either be proven, or even have their hypotheses shown to be
inconsistent, automatically via QE.
However, economists who this technology could benefit are usually unfamiliar
with QE, and the use of mathematical software generally. This motivated the
development of a Mathematica Package TheoryGuru, whose purpose is to lower the
costs of applying QE to economics. We describe the package's functionality and
give examples of its use.Comment: To appear in Proc ICMS 201
Non-linear Real Arithmetic Benchmarks derived from Automated Reasoning in Economics
We consider problems originating in economics that may be solved
automatically using mathematical software. We present and make freely available
a new benchmark set of such problems. The problems have been shown to fall
within the framework of non-linear real arithmetic, and so are in theory
soluble via Quantifier Elimination (QE) technology as usually implemented in
computer algebra systems. Further, they all can be phrased in prenex normal
form with only existential quantifiers and so are also admissible to those
Satisfiability Module Theory (SMT) solvers that support the QF_NRA. There is a
great body of work considering QE and SMT application in science and
engineering, but we demonstrate here that there is potential for this
technology also in the social sciences.Comment: To appear in Proc. SC-Square 2018. Dataset described is hosted by
Zenodo at: https://doi.org/10.5281/zenodo.1226892 . arXiv admin note:
substantial text overlap with arXiv:1804.1003
Bridging symbolic computation and economics: a dynamic and interactive tool to analyze the price elasticity of supply
It is not possible to achieve the objectives and skills of a program in economics, at the secondary and undergraduate levels, without resorting to graphic illustrations. In this way, the use of educational software has been increasingly recognized as a useful tool to promote students' motivation to deal with, and understand, new economic concepts. Current digital technology allows students to work with a large number and variety of graphics in an interactive way, complementing the theoretical results and the so often used paper and pencil calculations. The computer algebra system Mathematica is a very powerful software that allows the implementation of many interactive visual applications. Thanks to the symbolic and numerical capabilities of Mathematica, these applications allow the user to interact with the graphical and analytical information in real time. However, Mathematica is a commercially distributed application which makes it difficult for teachers and students to access. The main goal of this paper is to present a new dynamic and interactive tool, created with Mathematica and available in the Computable Document Format. This format allows anyone with a computer to use, at no cost, the PES(Linear)-Tool, even without an active Wolfram Mathematica license. The PES(Linear)-Tool can be used as an active learning tool to promote better student activity and engagement in the learning process, among students enrolled in socio-economic programs. This tool is very intuitive to use which makes it suitable for less experienced users.Funding Agency
Portuguese Foundation for Science and Technology
UID/ECO/04007/2019info:eu-repo/semantics/publishedVersio
Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets
In the context of strategic games, we provide an axiomatic proof of the
statement Common knowledge of rationality implies that the players will choose
only strategies that survive the iterated elimination of strictly dominated
strategies. Rationality here means playing only strategies one believes to be
best responses. This involves looking at two formal languages. One is
first-order, and is used to formalise optimality conditions, like avoiding
strictly dominated strategies, or playing a best response. The other is a modal
fixpoint language with expressions for optimality, rationality and belief.
Fixpoints are used to form expressions for common belief and for iterated
elimination of non-optimal strategies.Comment: 16 pages, Proc. 11th International Workshop on Computational Logic in
Multi-Agent Systems (CLIMA XI). To appea
Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition
There has been recent interest in the use of machine learning (ML) approaches
within mathematical software to make choices that impact on the computing
performance without affecting the mathematical correctness of the result. We
address the problem of selecting the variable ordering for cylindrical
algebraic decomposition (CAD), an important algorithm in Symbolic Computation.
Prior work to apply ML on this problem implemented a Support Vector Machine
(SVM) to select between three existing human-made heuristics, which did better
than anyone heuristic alone. The present work extends to have ML select the
variable ordering directly, and to try a wider variety of ML techniques.
We experimented with the NLSAT dataset and the Regular Chains Library CAD
function for Maple 2018. For each problem, the variable ordering leading to the
shortest computing time was selected as the target class for ML. Features were
generated from the polynomial input and used to train the following ML models:
k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision
tree (DT) and SVM, as implemented in the Python scikit-learn package. We also
compared these with the two leading human constructed heuristics for the
problem: Brown's heuristic and sotd. On this dataset all of the ML approaches
outperformed the human made heuristics, some by a large margin.Comment: Accepted into CICM 201
Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds
Given a formula in quantifier-free Presburger arithmetic, if it has a
satisfying solution, there is one whose size, measured in bits, is polynomially
bounded in the size of the formula. In this paper, we consider a special class
of quantifier-free Presburger formulas in which most linear constraints are
difference (separation) constraints, and the non-difference constraints are
sparse. This class has been observed to commonly occur in software
verification. We derive a new solution bound in terms of parameters
characterizing the sparseness of linear constraints and the number of
non-difference constraints, in addition to traditional measures of formula
size. In particular, we show that the number of bits needed per integer
variable is linear in the number of non-difference constraints and logarithmic
in the number and size of non-zero coefficients in them, but is otherwise
independent of the total number of linear constraints in the formula. The
derived bound can be used in a decision procedure based on instantiating
integer variables over a finite domain and translating the input
quantifier-free Presburger formula to an equi-satisfiable Boolean formula,
which is then checked using a Boolean satisfiability solver. In addition to our
main theoretical result, we discuss several optimizations for deriving tighter
bounds in practice. Empirical evidence indicates that our decision procedure
can greatly outperform other decision procedures.Comment: 26 page
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